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Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
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Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier .com/locate /cnsns
Effects of partial slip, viscous dissipation and Joule heatingon Von Kármán flow and heat transfer of an electricallyconducting non-Newtonian fluid
Bikash Sahoo *
Department of Mathematics, National Institute of Technology, Rourkela, Rourkela 769008, India
a r t i c l e i n f o a b s t r a c t
Article history:Received 19 June 2008Received in revised form 23 October 2008Accepted 23 October 2008Available online 11 November 2008
PACS:47.1547.5047.65
Keywords:Reiner–Rivlin fluidMHD flowRotating diskPartial slipHeat transferViscous dissipationJoule heatingFinite difference methodBroyden’s method
1007-5704/$ - see front matter � 2008 Elsevier B.Vdoi:10.1016/j.cnsns.2008.10.021
* Tel.: +91 0661 246 2706.E-mail address: [email protected]
The steady Von Kármán flow and heat transfer of an electrically conducting non-Newto-nian fluid is extended to the case where the disk surface admits partial slip. The fluid issubjected to an external uniform magnetic field perpendicular to the plane of the disk.The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Riv-lin fluid. The momentum equations give rise to highly non-linear boundary value problem.Numerical solutions for the governing non-linear equations are obtained over the entirerange of the physical parameters. The effects of slip, magnetic parameter and non-Newto-nian fluid characteristics on the velocity and temperature fields are discussed in detail andshown graphically. Emphasis has been laid to study the effects of viscous dissipation andJoule heating on the thermal boundary layer. It is interesting to find that the non-Newto-nian cross-viscous parameter has an opposite effect to that of the slip and the magneticparameter on the velocity and the temperature fields.
� 2008 Elsevier B.V. All rights reserved.
1. Introduction
The flow of a viscous fluid which arises due to the rotation of a disk in an otherwise stagnant fluid constitutes a prototypeof the three-dimensional boundary layer problems. It is seldom that one is able to find a closed form solution to the Navier–Stokes equations. The steady laminar flow of a viscous, incompressible fluid near a rotating disk is one of the few problems influid dynamics for which the Navier–Stokes equations admit an exact solution. The problem of disk flows has occupied acentral position in the field of fluid dynamics. Disk flows have immediate technical applications in rotating machinery, heatand mass exchangers, biomechanics and oceanography. A disk of infinite extent is in the plane z ¼ 0 and rotates with anangular velocity X about the z-axis (see Fig. 1) in an otherwise stagnant fluid. A boundary layer on each side of the disk devel-ops, through which the tangential component of the velocity is sheared from the value Xr at the surface to the value zero inthe free stream outside the boundary layer. In practice, only the flow on one side of the disk (z > 0) needs to be discussed. Thecentrifugal forces created by the rotating disk cause a radial outflow of the fluid within the boundary layer. Since, the radial
. All rights reserved.
z
V
roI
B
w
u
Fig. 1. Schematic representation of the flow domain.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2983
component of the velocity is zero both on the disk (‘no-slip’ condition) and in the free stream, external fluid is entrained axi-ally in to the boundary layer and ejected centrifugally again. This is therefore a fully three-dimensional flow which acts as apump. The radial flow is often referred to as the ‘free-disk pumping effect’.
The theoretical study of the flow near a rotating disk can be traced back to the Von Kármán’s similarity analysis [1]. That iswhy the flow is widely known as the Von Kármán flow. He devised an elegant similarity transformation which transformsthe axisymmetric Navier–Stokes equations into a set of ordinary differential equations. There are few minor inaccuracies inKármán’s analysis, which were corrected by Cochran [2]. Stuart [3] studied the effects of uniform suction on the flow due to arotating disk. Benton [4] improved the steady state solutions given by Cochran [2] and extended the problem to the transientstate. The most accurate solution so far seems to have been reported by Ackroyed [5]. Due to its fundamental nature in com-bination with its practical importance, a vast number of modifications and extensions of Von Kármán’s swirling flow prob-lem exists. The classical problem has been generalized in several manners to include diverse physical effects. The heattransfer aspects have been considered by Millsaps and Pohlhausen [6] for variety of Prandtl numbers in the steady state.Sparrow and Gregg [7] studied the steady state heat transfer from a rotating disk maintained at a constant temperatureto fluids at any Prandtl number. The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating diskhas been considered by Rogers and Lance [8]. Later, Evans [9] has extended the study with uniform suction at the surface ofthe disk. The effects of uniform blowing on the flow induced by the rotating disk was studied by Kuiken [10]. The compre-hensive review and detailed discussions by Owen and Rogers [11] gives an excellent overview of the flow phenomena occur-ring due to a single and two-disks systems. Attia [12] has adopted the Crank–Nicolson implicit method to find the solution ofthe system of non-linear differential equations arising due to the unsteady flow and heat transfer of a viscous incompressiblefluid.
Technical applications of rotating disk problems can be found for instance in viscometry, spin-coating, manufacturing anduse of computer disks, and in various rotating machinery components. In view of its wide applications in industrial and othertechnological fields, the problem of flow near a rotating disk has been extended to hydromagnetics. One of the major appli-cations of the MHD rotating disk flow is the manufacture of the magnetic hard disk drives (HDD) with high information stor-age capacity [13]. The hydromagnetic flow due to a rotating disk was first investigated by Katukani [14]. Sparrow and Cess[15] have studied the MHD flow due to a rotating disk by incorporating the energy equation. In view of its immense appli-cations, the flow problem continues to be studied till recently. Another reason for such an extensive study of the MHD flowdue to a rotating disk may be due to the inherent problems in numerical integration of MHD equations, which typically leadto a general solution that includes two solutions, one exponentially growing and the other exponentially decaying. It wouldnot be surprising that if the shooting techniques are employed, the actual numerical solution might be swamped by the par-asitic solutions, no matter whether the integration is carried forward or backward. For this reason, apparently special mea-sures need to be taken in numerical integration of the differential equations governing the flows in MHD, specially when the
2984 B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
strength of the magnetic field is not small and the flow is in infinite domain. Some interesting effects of the magnetic field onthe steady and unsteady flows due to the rotation of a disk of infinite or finite extent have been examined by various authors[16–18].
In all of the above studies the fluid is assumed to be Newtonian. In recent years it has generally been recognized that inindustrial applications non-Newtonian fluids are more appropriate than Newtonian fluids. That non-Newtonian fluids arefinding increasing applications in industries has given impetus to many researchers. The Von Kármán flow of different kindsof non-Newtonian fluids have been studied by Srivastava [19], Jain [20], Mithal [21], Rathna [22], Elliott [23] and Ji et al. [24]including diverse physical effects. A detailed discussion up to 1991 regarding the flow of non-Newtonian fluids due to rotat-ing disks can be found in the review paper by Rajagopal [25]. Ariel [26–28] has considered the flow of viscoelastic Walters ’B’and second grade fluids near a rotating disk. The Von Kármán flow of a non-Newtonian power-law fluid without and with atransverse magnetic field can be found in [29,30]. Recently, Attia [31–33] has considered the steady and unsteady VonKármán flow and heat transfer of Reiner–Rivlin fluid with suction or injection at the surface of the disk.
Moreover, it is evident that [11] the flow arising due to the rotation of the single disk closely resembles the flow near therotor in the rotor–stator system which has again vast industrial applications. Sahoo and Sharma [34] have studied the flow ofa non-Newtonian second grade fluid in a rotor–stator system. The issue of paucity of boundary conditions has been ad-dressed, and a numerical method has been adopted, which treats the higher order terms in the equations as lower iterates,essentially once again lowering the order of the equations.
In all the above mentioned studies, no attention has been given to the effect of partial slip on the flow due to a rotatingdisk. A completely different extension of Von Kármán’s one disk problem is the analysis of Sparrow et al. [35]. They consid-ered the flow of a Newtonian fluid due to the rotation of a porous surfaced disk and for that purpose replaced the conven-tional no-slip boundary conditions at the disk surface with a set of linear slip flow conditions. A substantial reduction intorque then occurred as a result of surface slip. A detailed account regarding the slip flow over an enclosed rotating diskcan be found in the doctoral dissertation of Sarafa [36]. In fact, the surface of the disk may be rough and not perfectly smoothas assumed. In that case, the no-slip boundary condition becomes impractical to apply exactly. If the characteristic scale ofthe roughness is small compared to the boundary layer thickness on the disk, the no-slip condition may be approximated bypartial slip condition applied to the envelop of the protuberances. The roughness may not be statistically isotropic. For exam-ple, it was found that for parallel, grooved surfaces the slip is larger in the direction along the grooves than the directiontransverse to the grooves [37]. The work of Miklavcic and Wang [38] takes into consideration of the influence of partial slipon the flow of a viscous fluid due to a rotating disk. The effects of slip over a rotating disk in a Newtonian fluid lubricated by anon-Newtonian fluid has been considered by Andersson and Rousselet [39]. Further, one can refer the works of Attia [40],Asghar et al. [41], Frusteri and Osalusi [42] and Osalusi et al. [43] pertaining to the Von Kármán slip flow with diverse phys-ical effects. Recently, Sahoo and Sharma [44] have investigated the effects of partial slip on the flow and heat transfer of anon-Newtonian Reiner–Rivlin fluid near a rotating disk. Their study reveals that the effects of slip is opposite to that of thenon-Newtonian fluid parameter on the flow and heat transfer due to a rotating disk.
It seems that there has been relatively little information regarding the influence of partial slip on the flow of a non-New-tonian fluid due to a rotating disk. Further, few notable findings regarding the Kármán flow with slip boundary conditionswhich prompted for the present investigation are as follows.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
−H
G
F
Fig. 2. Velocity profile for the Newtonian flow at Mn ¼ 0 and kð¼ gÞ ¼ 0.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2985
For the no-slip case (Kármán’s original problem [1]) the radial velocity starts from zero and reaches a maximum nearf ¼ 0:92, then decays to zero. But with partial slip Miklavcic and Wang [38] have reported that the maximum radial velocitydecreases and its location moves towards the disk. They have also found that a cross over far away from the disk for the ra-dial component of the velocity. The aforementioned findings are in agreement with those reported by Sahoo and Sharma [44]for non-Newtonian Reiner–Rivlin fluid in absence of magnetic field. Arikoglu and Ozkol [45] have observed that both the slipfactor and the magnetic flux decrease the velocity in all directions and thicken the thermal boundary layer. Andersson andRousselet [39] have found that the three-dimensional flow field is dramatically affected by accentuated velocity slip. Theyhave reported that the axial flow towards the disk, i.e. the pumping efficiency and the torque required to maintain steadyrotation of the disk, decrease monotonically with increasing slip. The MHD slip flow of a viscous Newtonian fluid over porousrotating disk, investigated by Frusteri and Osalusi [42] reveals that, the radial and the tangential velocity profiles are reducedby both slip coefficient and the magnetic field, whereas the axial velocity increases with an increase in the slip coefficient,contrary to the findings of [38,39,44].
Keeping the above curious findings in mind, the present work is devoted to examine the influence of partial slip, viscousdissipation and Joule heating on the steady Von Kármán flow and heat transfer of an electrically conducting non-NewtonianReiner–Rivlin fluid. Emphasis has been given to the combined effects of the non-Newtonian fluid parameter, magneticparameter and partial slip factor on the velocity and temperature fields. To the best of the author’s knowledge, no attentionhas been given to the effects of slip on MHD boundary layer flow and heat transfer with viscous dissipation and Joule heatingof a non-Newtonian Reiner–Rivlin fluid near a rotating disk. The obtained results have promising applications in engineering.This problem is not only important because of its technological significance but also in view of the interesting mathematicalfeatures presented by the equations governing the MHD slip flow and heat transfer.
2. Formulation of the problem
We consider an electrically conducting non-Newtonian Reiner–Rivlin fluid whose rheological behavior is governed bystress–strain rate law [46,47],
sij ¼ �pdi
j þ 2leij þ 2lcei
kekj ; ej
j ¼ 0; ð1Þ
where p denotes the pressure, l is the coefficient of viscosity, and lc is the coefficient of cross-viscosity. The fluid occupiesthe space z > 0 over an infinite rotating disk coinciding with the plane z ¼ 0 (see Fig. 1). The disk is assumed to be rotatingabout z-axis with an uniform angular velocity X. The fluid adheres to the surface of the disk partially and thus, motion of thefluid exhibits the slip condition. An external uniform magnetic field is applied perpendicular to the plane of the disk and hasa constant magnetic flux density B0. The Lorentz force, included in the momentum equation can be simplified if the followingassumptions are made:
� All physical quantities are constant.� The induced magnetic field is small compared with the applied magnetic field.� The electrical field is assumed to be zero.
These assumptions are valid when the magnetic Reynolds number is small and there is no displacement current Ref.[48]. It is natural to describe the flow in the cylindrical polar coordinates ðr;/; zÞ. In view of the rotational symmetry,@@/ � 0. The surface of the rotating disk is maintained at a uniform temperature Tw and far away from the wall, the freestream is kept at a constant temperature T1. Taking V ¼ ðu;v;wÞ for the steady flow,the equations of continuity andmotion are,
rþ @[email protected]¼ 0; ð2Þ
and
q [email protected]@rþw
@[email protected]� v2
r
� �þ rB2
0u ¼ @srr
@rþ @s
zr
@zþ
srr � s/
/
r; ð3Þ
q [email protected]@rþw
@[email protected]þ uv
r
� �þ rB2
0v ¼@sr
/
@rþ@sz
/
@zþ
2sr/
r; ð4Þ
q [email protected]@rþw
� �¼ @s
rz
@rþ @s
zz
@zþ sr
z
r: ð5Þ
The no-slip boundary conditions for the velocity field are given as
z ¼ 0; u ¼ 0; v ¼ rX; w ¼ 0; ð6aÞz!1; u! 0; v ! 0; p! p1: ð6bÞ
2986 B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
By using the Von Kármán [1] transformations
u ¼ rXFðfÞ; v ¼ rXGðfÞ; w ¼ffiffiffiffiffiffiffiXmp
HðfÞ; z ¼ffiffiffiffimX
rf; p� p1 ¼ �qmXP: ð7Þ
Eqs. (2)–(5) take the form
dHdfþ 2F ¼ 0; ð8Þ
d2F
df2 � HdFdf� F2 þ G2 �MnF � 1
2L
dFdf
� �2
� 3dGdf
� �2
� 2Fd2F
df2
" #¼ 0; ð9Þ
d2G
df2 � HdGdf� 2FG�MnGþ L
dFdf
dGdfþ F
d2G
df2
!¼ 0; ð10Þ
d2H
df2 � HdHdf� 7
2L
dHdf
d2H
df2 þdPdf¼ 0; ð11Þ
where f is the non-dimensional distance measured along the axis of rotation, F,G,H and P are non-dimensional functions of f,q is the density, m is the kinematic viscosity m ¼ l
q
� �and r is the electrical conductivity of the fluid. We define the magnetic
interaction number Mn by Mn ¼ rB20
qX . The boundary conditions (6) become,
f ¼ 0 : F ¼ 0; G ¼ 1; H ¼ 0; ð12aÞf!1 : F ! 0; G! 0; P ! 0; ð12bÞ
where L ¼ lcXl is the parameter that describes the non-Newtonian characteristic of the fluid. The above system (8)–(10) with
the prescribed boundary conditions (12) are sufficient to solve for the three components of the flow velocity. Eq. (11) can beused to solve for the pressure distribution at any point if required.
A generalization of Navier’s partial slip condition [38,49] gives, in the radial direction,
ujz¼0 ¼ k1szr jz¼0; ð13Þ
and in the azimuthal direction
vjz¼0 ¼ k2sz/jz¼0; ð14Þ
where k1, k2 are, respectively the slip coefficients. Let
k ¼ k1
ffiffiffiffiXm
rl g ¼ k2
ffiffiffiffiXm
rl: ð15Þ
With the help of transformations (7) and Eq. (15), the partial slip boundary conditions (13) and (14) become
Fð0Þ ¼ k½F 0ð0Þ � LFð0ÞF 0ð0Þ�; Gð0Þ � 1 ¼ g½G0ð0Þ � 2LFð0ÞG0ð0Þ�; Hð0Þ ¼ 0; ð16aÞFð1Þ ! 0; Gð1Þ ! 0: ð16bÞ
The governing equations are still Eqs. (8)–(10). The boundary conditions at infinity are Eq. (12b), but those on the disk arereplaced by Eq. (16a). Theses boundary conditions imply that both radial (F), and tangential (G) components of velocity van-ish sufficiently far away from the rotating disk, whereas the axial component of velocity (H) is anticipated to approach a yetunknown asymptotic limit for sufficiently large f value.
3. Heat transfer analysis
Due to the temperature difference between the surface of the disk and the ambient fluid, heat transfer takes place. Theenergy equation, with viscous dissipation and Joule heating takes the form,
qcp [email protected]@rþw
� �¼ j
1r@
� �þ @
( )þ l @u
@z
� �2
� �2" #
þ rB20ðu2 þ v2Þ: ð17Þ
Introducing the non-dimensional variable hðfÞ ¼ T�T1Tw�T1
and using the Von Kármán transformations (7), (17) becomes,
Hdhdf¼ 1
Pr
d2h
df2 þ EcðF 02 þ G02Þ þMnEcðF2 þ G2Þ; ð18Þ
where Tw is the temperature at the surface of the disk, T1 is the temperature of the ambient fluid at large distance from thedisk, Pr ¼ lcp
j is the Prandtl number and Ec ¼ r2X2
ðTw�T1Þcpis the Eckert number. The boundary conditions in terms of the non-
dimensional parameter h are expressed as
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2987
f ¼ 0 : h ¼ 1;f!1 : h! 0: ð19Þ
The heat transfer from the disk surface to the fluid is computed by the application of the Fourier’s law, q ¼ �j @[email protected]
� �w. Intro-
ducing the transformed variables, the expression for q becomes
q ¼ �jðTw � T1ÞffiffiffiffiXm
rdhð0Þ
df: ð20Þ
By rephrasing the heat transfer results in terms of the Nusselt number defined as Nu ¼qffiffimX
pjðTw�T1Þ, we get
Nu ¼ �dhð0Þ
df: ð21Þ
The action of the viscosity in the fluid adjacent to the disk tends to set up tangential shear stress �su, which opposes the rota-tion of the disk. There is also a surface shear stress �sr in the radial direction. In terms of the variables of the analysis, theexpressions of �su and �sr are respectively given as
�s/
qrffiffiffiffiffiffiffiffiffimX3
p ¼ s/ ¼dGð0Þ
df� 2LFð0Þ dGð0Þ
df;
�sr
qrffiffiffiffiffiffiffiffiffimX3
p ¼ sr ¼dFð0Þ
df� LFð0ÞdFð0Þ
df: ð22Þ
Although the above calculation is strictly speaking applicable to an infinite disk only, we may utilize the same results for afinite disk, provided the radius r is large enough. We shall now evaluate the most significant physical quantity of interest, theturning moment (or torque) for the disk with fluid on both sides. The dimensionless form of the moment coefficient is givenby
Cm ¼�2p½G0ð0Þ � 2LFð0ÞG0ð0Þ�ffiffiffiffiffiffi
Rep ; ð23Þ
where Re ¼ r2Xm is the local rotational Reynolds number based on the radius and the tip velocity.
4. Numerical solution of the problem
The system of non-linear differential Eqs. (8)–(10) and (18) is solved under the boundary conditions (16) and (19), respec-tively. One can see that the initial boundary conditions for F and G in Eq. (16a) are unknown contrary to the case of no-slipboundary conditions (12a). Hence, the solution of the system can not proceed numerically using any standard integrationroutine. Here, we have adopted a second order numerical technique, similar to that of used in [44,50], which combinesthe features of the finite difference method and the shooting method. The method is accurate because it uses central differ-ences. The semi-infinite integration domain f 2 ½0;1Þ is replaced by a finite domain f 2 ½0; f1Þ. In practice, f1 should be cho-sen sufficiently large so that the numerical solution closely approximates the terminal boundary conditions (16b). If f1 is notlarge enough, the numerical solution will not only depend on the physical parameters L, Mn, k, but also on f1. Hence, a finitevalue, large enough, has been substituted for f1, the numerical infinity, to ensure that the solutions are not affected byimposing the asymptotic conditions at a finite distance. The value of f1 has been kept invariant during the run of the pro-gram. The value of f1 ¼ 10:0 is found to be adequate to simulate f ¼ 1 for all the cases shown in Figs. 3–22. However, forhigher values of the flow parameters, the numerical integrations are performed over substantially larger domain to ensurethat the outer boundary conditions at f1 are satisfied.
Now suppose we introduce a mesh defined by
fi ¼ ihði ¼ 0;1; . . . ; nÞ; ð24Þ
h being the mesh size, and discretize Eqs. (8)–(10) and (18) using the central difference approximations for the derivatives,then the following equations are obtained.
Fiþ1 � 2Fi þ Fi�1
h2 � HiFiþ1 � Fi�1
2h
� �� F2
i þ G2i �MnFi
� 12
LFiþ1 � Fi�1
2h
� �2
� 3Giþ1 � Gi�1
2h
� �2
� 2FiFiþ1 � 2Fi þ Fi�1
h2
� �" #¼ 0; ð25Þ
Giþ1 � 2Gi þ Gi�1
h2 � HiGiþ1 � Gi�1
2h
� �� 2FiGi �MnGi
þ LFiþ1 � Fi�1
2h
� �Giþ1 � Gi�1
2h
� �þ Fi
Giþ1 � 2Gi þ Gi�1
h2
� � ¼ 0; ð26Þ
2988 B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
Hihiþ1 � hi�1
2h
� �� 1
Pr
hiþ1 � 2hi þ hi�1
h2
� �
� EcFiþ1 � Fi�1
2h
� �2
þ Giþ1 � Gi�1
2h
� �2" #
�MnEcðF2i þ G2
i Þ ¼ 0; ð27Þ
Hiþ1 ¼ Hi �h2ðFi þ Fiþ1Þ: ð28Þ
Note that Eqs. (9), (10) and (18), which are written at ith mesh point, the first and second derivatives are approximated bythe central differences centered at ith mesh point, while in Eq. (8), which is written at iþ 1
2
� �th mesh point, the first deriv-
ative is approximated by the difference quotient at ith and ðiþ 1Þth mesh points, and the right hand sides are approximatedby the respective averages at the same two mesh points. This scheme ensures that the accuracy of Oðh2Þ is preserved in thediscretization.
Eqs. (25)–(27) are three term recurrence relations in F,G and h, respectively. Hence, in order to start the recursion, besidesthe values of F0,G0 and h0, the values of F1,G1 and h1 are also required. These values can be obtained by Taylor series expan-sion near f ¼ 0 for F,G and h.
If
F 0ð0Þ ¼ s1; G0ð0Þ ¼ s2 and h0ð0Þ ¼ s3; ð29Þ
we have
F1 ¼ Fð0Þ þ hF 0ð0Þ þ h2
2F 00ð0Þ þ Oðh2Þ
G1 ¼ Gð0Þ þ hG0ð0Þ þ h2
2G00ð0Þ þ Oðh2Þ
h1 ¼ hð0Þ þ hh0ð0Þ þ h2
2h00ð0Þ þ Oðh2Þ ð30Þ
The values Fð0Þ, Gð0Þ and hð0Þ are given as boundary conditions in (16) and (19). The values F 00ð0Þ, G00ð0Þ and h00ð0Þ can beobtained directly from (9), (10) and (18) and using the values in (29). After obtaining the values of F1, G1 and h1, the inte-gration can now be performed as follows. H1 can be obtained from (28). Using the values of H1 in (25)–(27), the values ofF2, G2 and h2 are obtained. At the next cycle, H2 is computed from (28) and is used in Eqs. (25)–(27) to obtain F3, G3 andh3, respectively. The order indicated above is followed for the subsequent cycles. The integration is carried out until the val-ues of F, G, H and h are obtained at all the mesh points.
Note that we need to satisfy the three asymptotic boundary conditions (16) and (19). In fact, s1, s2 and s3 are found byshooting method along with fourth order Runge–Kutta method so as to fulfil the free boundary conditions at f ¼ f1 in(16) and (19). The guesses on F 0ð0Þ, G0ð0Þ and h0ð0Þ can be improved by a suitable zero-finding algorithm. Here, amongst sev-eral choices, one can apply a variation of secant method or use Newton’s method as the zero-finding algorithm. In Newton’smethod, one has to solve sixteen equations at each integration step (four original Eqs. (25)–(28) plus four equations resultingfrom taking partial derivatives with respect to each unknown s1, s2 and s3). Again, the three-dimensional version of the se-cant method is quite sensitive to the initial guesses and require a lot more iterations. However, we found Broyden’s [51,52]method quite adequate. The fact that the algorithm has an accuracy of only Oðh2Þ need not concern us unduly, as we caneasily hike the accuracy to Oðh4Þ by invoking Richardson’s extrapolation. With reasonably close trial values to start the iter-ations, the convergence to the actual values within an accuracy of Oð10�6Þ could be attained in 9–12 iterations.
Note that even though we have used finite difference scheme to approximate the derivatives, we are still using the shoot-ing method to solve the present boundary value problem. In fact, the shooting method is straight forward and it works wellfor small values of the flow parameters, whereas a special merit of the algorithm reported in the present work is that it isapplicable for arbitrary values of the flow parameters L, Mn and k (or g).
It is customary to mention that the above solution procedure belongs to one of the major classes of solution strategies forBVPs Refs. [53,54].
5. Results and discussion
The method described above was translated into a FORTRAN 90 program and was run on a pentium IV personal computer.The value of f1, the numerical infinity has been taken large enough and kept invariant through out the run of the program.To see if the program runs correctly, the results of F 0ð0Þ, �G0ð0Þ and Fmax for the viscous fluid (L ¼ 0) subject to no-slip bound-ary conditions (k ¼ g ¼ 0), are compared (see Table 1) with the exact solutions reported by Ariel [17], for different values ofthe magnetic parameter. The comparison is found to be in good agreement. In order to have an insight of the flow and heattransfer characteristics, results are plotted graphically in figures for the uniform roughness (k ¼ g), different choice of thenon-dimensional magnetic and non-Newtonian parameters. Moreover, the values of the important standard parameters like�H1, sr , s/, Cm and Nu have been tabulated in Table 2, for different values of the flow parameters when Pr ¼ 1:0, Ec ¼ 0:3 andRe ¼ 4:0.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2989
Fig. 2 represents the corresponding velocity profile for the no-slip Newtonian (L ¼ 0) flow at Mn ¼ 0. It is clear that inabsence of slip, the radial velocity F starts from zero and reaches a maximum near f ¼ 0:92 and then decays to zero.
Table 1Variations of F 0ð0Þ, �G0ð0Þ and Fmax with Mn for L ¼ 0 and kð¼ gÞ ¼ 0.
Mn F0ð0Þ �G0ð0Þ Fmax
Current result Ariel [17] Current result Ariel [17] Current result Ariel [17]
0.0 0.510214 0.510233 0.615909 0.615922 0.180740 0.1807670.2 0.453137 0.453141 0.708794 0.708795 0.146343 0.1463610.4 0.405575 0.405576 0.802376 0.802376 0.119166 0.1192540.6 0.366698 0.366698 0.894475 0.894476 0.098445 0.0985370.8 0.335090 0.335090 0.983607 0.983607 0.082841 0.0828421.0 0.309259 0.309258 1.069053 1.069053 0.070756 0.07087310.0 0.105310 0.105310 3.164907 3.164907 0.008312 0.00831612.0 0.096163 – 3.466103 – 0.006923 –14.0 0.089045 – 3.743246 – 0.005924 –16.0 0.083303 – 4.001301 – 0.005203 –18.0 0.078545 – 4.243731 – 0.004623 –20.0 0.074518 0.074518 4.473067 4.473067 0.004164 –50.0 0.047139 0.047139 7.071303 7.071303 0.001650 –100 0.033334 0.033333 10.000083 10.000083 0.000833 –
Table 2Variations of �H1 , sr , s/ , Cm , and Nu with different flow parameters.
L Mn kð¼ gÞ �H1 sr s/ Cm Nu
0.0 0.137633 0.037059 �0.513095 1.611937 0.0932985.0 1.053198 0.152564 �0.205124 0.644416 0.33298110.0 1.0 1.0 1.453484 0.091378 �0.098222 0.308573 0.30443815.0 1.731162 0.063418 �0.065318 0.205204 0.28306920.0 1.963864 0.048376 �0.049116 0.154302 0.266871
0.0 1.158215 0.188433 �0.313511 0.984923 0.4789835.0 0.123879 0.075839 �0.642264 2.017733 0.074471
2.0 10.0 1.0 0.053750 0.049252 �0.731521 2.298140 0.05621915.0 0.031972 0.036892 �0.775719 2.436993 0.05584920.0 0.021903 0.029676 �0.803170 2.523233 0.057714
0.0 1.266822 0.923505 �0.914327 2.872442 0.36516210.0 0.000906 0.000094 �0.096926 0.304502 0.099538
2.0 1.0 20.0 0.000083 0.000006 �0.049447 0.155342 0.09987730.0 0.000020 0.000001 �0.033132 0.104088 0.09995040.0 0.000007 0.000000 �0.024902 0.078231 0.099974
0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
ζ
F(ζ
)
L=0L=2L=4L=6
Fig. 3. Variation of F with L at Mn ¼ 1:0 and kð¼ gÞ ¼ 2:0.
0 2 4 6 8 10−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
ζ
F(ζ
)
λ=η=0=2.0=4.0=6.0
Fig. 6. Variation of F with kð¼ gÞ at L ¼ 2:0 and Mn ¼ 2:0.
0 2 4 6 8 10−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
ζ
F(ζ
)
Mn=0
Mn=0.5
Mn=1.0
Mn=1.5
Fig. 4. Variation of F with Mn at L ¼ 2:0 and kð¼ gÞ ¼ 2:0.
0 2 4 6 8 10−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
ζ
F(ζ
)
λ=η=0=2.0=4.0=6.0
Fig. 5. Variation of F with kð¼ gÞ at L ¼ 2:0 and Mn ¼ 0.
2990 B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
G(ζ
)
λ=η=0=2.0=4.0=6.0
Fig. 9. Variation of G with kð¼ gÞ at L ¼ 2:0 and Mn ¼ 2:0.
0 2 4 6 8 10−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ζ
G(ζ
)
L=0L=2L=4L=6
Fig. 7. Variation of G with L at Mn ¼ 1:0 and kð¼ gÞ ¼ 2:0.
0 2 4 6 8 10−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ζ
G(ζ
)
Mn=0.0
Mn=0.5
Mn=1.0
Mn=1.5
Fig. 8. Variation of G with Mn at L ¼ 2:0 and kð¼ gÞ ¼ 2:0.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2991
0 2 4 6 8 10−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ζ
−H
(ζ)
λ=η=0=2.0=4.0=6.0
Fig. 12. Variation of �H with kð¼ gÞ at L ¼ 2:0, Mn ¼ 2:0.
0 2 4 6 8 10−0.2
0
0.2
0.4
0.6
0.8
1
1.2
ζ
H(
ζ)
L=0L=2L=4L=6
Fig. 10. Variation of �H with L at Mn ¼ 1:0 and kð¼ gÞ ¼ 2:0.
0 2 4 6 8 10−0.2
0
0.2
0.4
0.6
0.8
1
1.2
ζ
−H
(ζ)
Mn=0.0
Mn=0.5
Mn=1.0
Mn=1.5
Fig. 11. Variation of �H with Mn at L ¼ 2:0 and kð¼ gÞ ¼ 2:0.
2992 B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
θ(ζ
)
λ=η=0=2.0=4.0=6.0
Fig. 15. Variation of h with kð¼ gÞ at L ¼ 2:0, Mn ¼ 0:1, Pr ¼ 2:0 and Ec ¼ 0:3.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
θ(ζ
)
Mn=0.0
Mn=0.5
Mn=1.0
Mn=1.5
Fig. 14. Variation of h with Mn at L ¼ 2:0, kð¼ gÞ ¼ 2:0, Pr ¼ 1:0 and Ec ¼ 0:3.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
θ(ζ
)
L=0L=2L=4L=6
Fig. 13. Variation of h with L at Mn ¼ 0:1, kð¼ gÞ ¼ 2:0, Pr ¼ 1:0 and Ec ¼ 0:3.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2993
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
ζ
θ(ζ
)
Ec=0.0
Ec=3.0
Ec=6.0
Ec=9.0
Fig. 18. Variation of h with Ec at L ¼ 2:0, Mn ¼ 0, kð¼ gÞ ¼ 2:0 and Pr ¼ 2:0.
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ζ
θ(ζ
)
Pr=1.0
Pr=3.0
Pr=5.0
Pr=7.0
Fig. 16. Variation of h with Pr at L ¼ 2:0, Mn ¼ 0:1, kð¼ gÞ ¼ 2:0 and Ec ¼ 0:3.
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ζ
θ(ζ
)
Ec=0.0
Ec=3.0
Ec=6.0
Ec=9.0
Fig. 17. Variation of h with Ec at L ¼ 2:0, Mn ¼ 1:0, kð¼ gÞ ¼ 2:0 and Pr ¼ 2:0.
2994 B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
0 5 10 15 200
0.5
1
1.5
2
2.5
3
L
Tr
λ=(η)=0
Fig. 19. Variation of sr with L at M ¼ 1:0 and kð¼ gÞ ¼ 0.
0 5 10 15 200.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
L
Tr
λ(=η)=1.0
Fig. 20. Variation of sr with L at M ¼ 1:0 and kð¼ gÞ ¼ 1:0.
0 5 10 15 20−3
−2.8
−2.6
−2.4
−2.2
−2
−1.8
−1.6
−1.4
−1.2
−1
L
Tφ
M=1.0λ(=η)=0
Fig. 21. Variation of s/ with L at M ¼ 1:0 and kð¼ gÞ ¼ 0.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2995
0 5 10 15 20−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
L
Tφ
M=1.0λ(=η)=2.0
Fig. 22. Variation of s/ with L at M ¼ 1:0 and kð¼ gÞ ¼ 2:0.
2996 B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998
Figs. 3–6 respectively depict the variations of the radial component of velocity FðfÞ as a function of f for different values ofthe non-Newtonian parameter L, magnetic parameter Mn, and the slip factor kð¼ gÞ. It is observed that (see Fig. 3) for a givenposition f, in presence of slip, FðfÞ increases with L, and hence, the maximum value of the radial component Fmax increases.From Fig. 4, it is clear that FðfÞ decreases with an increase in the magnetic parameter Mn due to the inhibiting influence of theLorentz force. Figs. 5 and 6 delineate that slip has a prominent effect on the radial component of the velocity near the disk. Asthe slip parameter increases in magnitude, permitting more fluid to slip past the disk, the maximum radial velocity decreasesand its location moves towards the disk. Fig. 5 depicts, in absence of the magnetic field, a prominent cross over of the curvesFðfÞ is observed near f ¼ 4:0, showing although slip decreases the radial component of velocity near the disk, it increase thevelocity far from the disk. This cross over was also observed by Miklavcic and Wang [38] for a viscous fluid; Sahoo and Shar-ma [44] for the non-Newtonian Reiner–Rivlin fluid without magnetic field. However, it is interesting to find that as the mag-netic parameter Mn increases, the position of the cross over shifts towards the f axis and finally coincides with it, as is clearfrom Fig. 6. In the limiting case as kð¼ gÞ ! 1, i.e. when the fluid is entirely potential, the rotating disk has no effect to rotatethe fluid particles, and therefore, the fluid becomes at rest.
In Figs. 7 and 8 we plot the dimensionless azimuthal component of velocity GðfÞ as a function of f with L and Mn, respec-tively. The figures show that the shear driven motion GðfÞ in the azimuthal direction decays rapidly with the distance f fromthe disk. Its value in general increases with an increase in the value of L, but decreases as the magnetic field is increased. Thatis Lorentz force, which opposes the flow leads to enhanced deceleration of the azimuthal velocity. Fig. 9 depicts that the sliphas a prominent effect on G near the disk and for a given position f, GðfÞ decreases with an increase in the slip factor kð¼ gÞ.
Figs. 10–12 show the axial velocity profiles�HðfÞwith different flow parameters. From the figures, it is clear that the axialcomponent of velocity increases with an increase in the non-Newtonian parameter L and decreases with the magneticparameter Mn. As pointed out earlier, due to the rotation of the disk, the layer near the surface is thrown outwards owingto the action of the centrifugal force. This is compensated by particles which flow in an axial direction (�H) towards the disk,to be in turn carried and ejected centrifugally. The gradual reduction of the peak in the F-profiles in Fig. 6 with increasingvalues of the slip parameter is reflected in the distributions of the axial velocity component in Fig. 12. The distinct inflectionpoint in the �H-profiles for higher values of kð¼ gÞ seems to gradually disappear. This is a consequence of the direct couplingbetween the radial and the axial velocity components through the continuity constraint (8). The reduction of the radialvelocity FðfÞ with increasing kð¼ gÞ automatically gives rise to a reduced axial inflow since,
�Hð1Þ ¼ 2Z 1
0Fdf: ð31Þ
The variation of the non-dimensional temperature profiles hðfÞ with different flow parameters are shown in Figs. 13–18.From Fig. 13 it is clear that for a given position f, the temperature decreases with an increase in the non-Newtonian param-eter L. On the other hand, the magnetic and the slip parameters have opposite effects on the temperature profile. Since themagnitude of the increase of thermal boundary layer thickness due to the magnetic parameter Mn is more appreciable thanthat decreased due to the non-Newtonian parameter L, we can expect that the thermal characteristics are more influenced byMn than those by L in this problem. In Fig. 15, one can observe that as the slip parameter increases, the temperature getsincreased, resulting in a increase in the thermal boundary layer thickness. Fig. 16 shows the variation of the temperaturewith the Prandtl number Pr . As was expected, h gets decreased with an increase in Pr , and thus, the thermal boundary layerthickness gets decreased. In fact, it is well known that the thermal boundary layer thickness is inversely proportional to thesquare root of Prandtl number. Hence, the decrease of temperature profile with Pr is straightforward.
B. Sahoo / Commun Nonlinear Sci Numer Simulat 14 (2009) 2982–2998 2997
The heat generated due to the Joule heating and viscous dissipation are characterized by Mn and Ec respectively. In Figs. 17and 18, we plot hðfÞ with the Eckert number Ec , when the other flow parameters are kept constant. The inclusion of Jouleheating Mn–0 has a prominent effect on the temperature distribution, as is clear from Fig. 17. The effects of Joule heatingon the thermal characteristics at the surface of the disk are magnified when Ec becomes large. In both of the aforementionedfigures, one can observe that there is a temperature overshoot near the disk, with an increase in the Eckert number. This is ofcourse a consequence of the fact that for higher values of the Eckert number, there is significant generation of heat due toviscous dissipation near the disk, so that the temperature in the region very close to the disk exceeds Tw.
Variations of the radial component of velocity at f1, (i.e.�H1), the non-dimensional radial shear stress (sr), the tangentialshear stress (s/), and the Nusselt number with different values of the flow parameters have been tabulated in Table 2. Theinflow rate at infinity, as one can see form this Table, decreases both with the slip factor and the magnetic interaction num-ber. This is quite natural since the radially outwards boundary layer is fed by the axial flow at infinity. The effect of the mag-netic field is to reduce and eventually suppress (Fig. 4) the radially directed out flow. An accompanying reduction of the axialflow (Fig. 11) towards the disk is observed together of the thinning of the boundary layer adjacent to the disk, there byincreasing the torque (Cm) (see: Table 2) required to maintain rotation of the disk at the prescribed angular velocity. It isobserved from Table 2 that an increase in the slip parameter substantially reduces the radial and the tangential shear stres-ses sr and s/. One of the novel findings (see Table 2 and Fig. 20) of the present investigation is that in presence of the slipfactor, the radial shear stress sr initially increases with the non-Newtonian parameter L, reaches its maximum value (criticalvalue not precisely determined), and then starts falling. However, such a turning point is not observed for the no-slip case(Fig. 19). Similarly, Figs. 21 and 22 reveal interesting effects of the slip on the tangential shear stress s/. Moreover, an in-crease in the magnetic interaction parameter Mn, surprisingly decreases the Nusselt number Nu, and hence, the heat transferrate from the surface of the disk to the ambient fluid up to certain value of Mn, and then the Nusselt number starts increasingwith Mn, as is clear from Table 2.
6. Conclusions
This work is a worthwhile attempt to study of the effects of partial slip, viscous dissipation and Joule heating on the flowand heat transfer of an electrically conducting non-Newtonian Reiner–Rivlin fluid due to a rotating disk. The new set of slipflow boundary conditions aimed to accommodate for the partial slip effect. An effective second order numerical scheme hasbeen adopted to solve the resulting system of highly non-linear differential equations subject to the slip boundary condi-tions. The use of Broyden’s method has indeed enhanced the efficiency of the present algorithm by reducing the computa-tional (CPU) time. The combined effects of the slip and the magnetic interaction parameter are studied in detail. It isinteresting to find that the non-Newtonian cross-viscous parameter L has an opposite effect to that of the slip and the mag-netic parameter on the velocity and the temperature fields. The inclusion of viscous dissipation and Joule heating have prom-inent effects on the thermal boundary layer. Moreover, for the slip flow, a turning point in curve presenting the non-dimensional radial shear stress (sr) has been found, which is absent for the no-slip case.
Acknowledgements
The suggestions made by the anonymous reviewers to improve the quality of the work are highly acknowledged.
References
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